We review recent determinations of the present-day mass function (PDMF) and initial mass function (IMF) in various components of the Galaxy—disk, spheroid, young, and globular clusters—and in conditions characteristic of early star formation. As a general feature, the IMF is found to depend weakly on the environment and to be well described by a power-law form forM , and a lognormal form below, except possibly for m!1 early star formation conditions. The disk IMF for single objects has a characteristic mass around M , m!0.08 c and a variance in logarithmic mass , whereas the IMF for multiple systems hasM , and . j!0.7 m!0.2 j!0.6 c The extension of the single MF into the brown dwarf regime is in good agreement with present estimates of L- and T-dwarf densities and yields a disk brown dwarf number density comparable to the stellar one, n!n! BD " pc !3 .T he IMF of young clusters is found to be consistent with the disk fi eld IMF, providing the same correction 0.1 for unresolved binaries, confirming the fact that young star clusters and disk field stars represent the same stellar population. Dynamical effects, yielding depletion of the lowest mass objects, are found to become consequential for ages!130 Myr. The spheroid IMF relies on much less robust grounds. The large metallicity spread in the local subdwarf photometric sample, in particular, remains puzzling. Recent observations suggest that there is a continuous kinematic shear between the thick-disk population, present in local samples, and the genuine spheroid one. This enables us to derive only an upper limit for the spheroid mass density and IMF. Within all the uncertainties, the latter is found to be similar to the one derived for globular clusters and is well represented also by a lognormal form with a characteristic mass slightly larger than for the disk, M , ,e xcluding as ignif icant population of m!0.2-0.3 c brown dwarfs in globular clusters and in the spheroid. The IMF characteristic of early star formation at large redshift remains undetermined, but different observational constraints suggest that it does not extend below!1M , .T hese results suggest a characteristic mass for star formation that decreases with time, from conditions prevailing at large redshift to conditions characteristic of the spheroid (or thick disk) to present-day conditions.Theseconclusions,however, remain speculative, given the large uncertainties in the spheroid and early star IMF determinations. These IMFs allow a reasonably robust determination of the Galactic present-day and initial stellar and brown dwarf contents. They also have important galactic implications beyond the Milky Way in yielding more accurate mass-to-light ratio determinations. The mass-to-light ratios obtained with the disk and the spheroid IMF yield values 1.8-1.4 times smaller than for a Salpeter IMF, respectively, in agreement with various recent dynamical determinations. This general IMF determination is examined in the context of star formation theory. None of the theories based on a Jeans-type mechanism, where fragmentation is due only to gravity, can fulfill all the observational constraints on star formation and predict a large number of substellar objects. On the other hand, recent numerical simulations of compressible turbulence, in particular in super-Alfvenic conditions, seem to reproduce both qualitatively and quantitatively the stellar and substellar IMF and thus provide an appealing theoretical foundation. In this picture, star formation is induced by the dissipation of large-scale turbulence to smaller scales through radiative MHD shocks, producing filamentary structures. These shocks produce local nonequilibrium structures with large density contrasts, which collapse eventually in gravitationally bound objects under the combined influence of turbulence and gravity. The concept of a single Jeans mass is replaced by a distribution of local Jeans masses, representative of the lognormal probability density function of the turbulent gas. Objects below the mean thermal Jeans mass still have a possibility to collapse, although with a decreasing probability.

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Galactic stellar and substellar initial mass function

We review current understanding of star formation, outlining an overall theoretical framework and the observations that motivate it. A conception of star formation has emerged in which turbulence plays a dual role, both creating overdensities to initiate gravitational contraction or collapse, and countering the effects of gravity in these overdense regions. The key dynamical processes involved in star formation—turbulence, magnetic fields, and self-gravity— are highly nonlinear and multidimensional. Physical arguments are used to identify and explain the features and scalings involved in star formation, and results from numerical simulations are used to quantify these effects. We divide star formation into large-scale and small-scale regimes and review each in turn. Large scales range from galaxies to giant molecular clouds (GMCs) and their substructures. Important problems include how GMCs form and evolve, what determines the star formation rate (SFR), and what determines the initial mass function (IMF). Small scales range from dense cores to the protostellar systems they beget. We discuss formation of both low- and high-mass stars, including ongoing accretion. The development of winds and outflows is increasingly well understood, as are the mechanisms governing angular momentum transport in disks. Although outstanding questions remain, the framework is now in place to build a comprehensive theory of star formation that will be tested by the next generation of telescopes.

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Theory of Star Formation

CO line emission represents the most accessible and widely used tracer of the molecular interstellar medium. This renders the translation of observed CO intensity into total H2 gas mass critical to understand star formation and the interstellar medium in our Galaxy and beyond. We review the theoretical underpinning, techniques, and results of efforts to estimate this CO-to-H2 “conversion factor,” XCO, in different environments. In the Milky Way disk, we recommend a conversion factor XCO = 2×10 20 cm −2 (K km s −1 ) −1 with ±30% uncertainty. Studies of other “normal galaxies” return similar values in Milky Way-like disks, but with greater scatter and systematic uncertainty. Departures from this Galactic conversion factor are both observed and expected. Dust-based determinations, theoretical arguments, and scaling relations all suggest that XCO increases with decreasing metallicity, turning up sharply below metallicity ≈ 1/3–1/2 solar in a manner consistent with model predictions that identify shielding as a key parameter. Based on spectral line modeling and dust observations, XCO appears to drop in the central, bright regions of some but not all galaxies, often coincident with regions of bright CO emission and high stellar surface density. This lower XCO is also present in the overwhelmingly molecular interstellar medium of starburst galaxies, where several lines of evidence point to a lower CO-to-H2 conversion factor. At high redshift, direct evidence regarding the conversion factor remains scarce; we review what is known based on dynamical modeling and other arguments. Subject headings: ISM: general — ISM: molecules — galaxies: ISM — radio lines: ISM

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The CO-to-H2 Conversion Factor

We present a comprehensive analysis of the relationship between star formation rate surface density, ΣSFR, and gas surface density, Σgas, at sub-kpc resolution in a sample of 18 nearby galaxies. We use high-resolution H I data from The H I Nearby Galaxy Survey, CO data from HERACLES and the BIMA Survey of Nearby Galaxies, 24 μm data from the Spitzer Space Telescope, and UV data from the Galaxy Evolution Explorer. We target seven spiral galaxies and 11 late-type/dwarf galaxies and investigate how the star formation law differs between the H2 dominated centers of spiral galaxies, their H I dominated outskirts and the H I rich late-type/dwarf galaxies. We find that a Schmidt-type power law with index N = 1.0 ± 0.2 relates ΣSFR and ΣH2 across our sample of spiral galaxies, i.e., that H2 forms stars at a constant efficiency in spirals. The average molecular gas depletion time is ~2 × 109 years. The range of ΣH2 over which we measure this relation is ~3-50 M ☉ pc–2, significantly lower than in starburst environments. We find the same results when performing a pixel-by-pixel analysis, averaging in radial bins, or when varying the star formation tracer used. We interpret the linear relation and constant depletion time as evidence that stars are forming in giant molecular clouds with approximately uniform properties and that ΣH2 may be more a measure of the filling fraction of giant molecular clouds than changing conditions in the molecular gas. The relationship between total gas surface density (Σgas) and ΣSFR varies dramatically among and within spiral galaxies. Most galaxies show little or no correlation between ΣHI and ΣSFR. As a result, the star formation efficiency (SFE), ΣSFR/Σgas, varies strongly across our sample and within individual galaxies. We show that this variation is systematic and consistent with the SFE being set by local environmental factors: in spirals the SFE is a clear function of radius, while the dwarf galaxies in our sample display SFEs similar to those found in the outer optical disks of the spirals. We attribute the similarity to common environments (low density, low metallicity, H I dominated) and argue that shear (which is typically absent in dwarfs) cannot drive the SFE. In addition to a molecular Schmidt law, the other general feature of our sample is a sharp saturation of H I surface densities at ΣHI ≈ 9 M ☉ pc–2 in both the spiral and dwarf galaxies. In the case of the spirals, we observe gas in excess of this limit to be molecular.

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The Star Formation Law in Nearby Galaxies on Sub-Kpc Scales

Understanding the formation of stars in galaxies is central to much of modern astrophysics. However, a quantitative prediction of the star formation rate and the initial distribution of stellar masses remains elusive. For several decades it has been thought that the star formation process is primarily controlled by the interplay between gravity and magnetostatic support, modulated by neutral-ion drift (known as ambipolar diffusion in astrophysics). Recently, however, both observational and numerical work has begun to suggest that supersonic turbulent flows rather than static magnetic fields control star formation. To some extent, this represents a return to ideas popular before the importance of magnetic fields to the interstellar gas was fully appreciated. This review gives a historical overview of the successes and problems of both the classical dynamical theory and the standard theory of magnetostatic support, from both observational and theoretical perspectives. The outline of a new theory relying on control by driven supersonic turbulence is then presented. Numerical models demonstrate that, although supersonic turbulence can provide global support, it nevertheless produces density enhancements that allow local collapse. Inefficient, isolated star formation is a hallmark of turbulent support, while efficient, clustered star formation occurs in its absence. The consequences of this theory are then explored for both local star formation and galactic-scale star formation. It suggests that individual star-forming cores are likely not quasistatic objects, but dynamically collapsing. Accretion onto these objects varies depending on the properties of the surrounding turbulent flow; numerical models agree with observations showing decreasing rates. The initial mass distribution of stars may also be determined by the turbulent flow. Molecular clouds appear to be transient objects forming and dissolving in the larger-scale turbulent flow, or else quickly collapsing into regions of violent star formation. Global star formation in galaxies appears to be controlled by the same balance between gravity and turbulence as small-scale star formation, although modulated by cooling and differential rotation. The dominant driving mechanism in star-forming regions of galaxies appears to be supernovae, while elsewhere coupling of rotation to the gas through magnetic fields or gravity may be important.

Control of star formation by supersonic turbulence

In this paper they conclude that turbulent fragmentation is unavoidable in super-sonically turbulent molecular clouds, and given the success of the present model to predict the observed shape of the Stellar IMF, they conclude that turbulent fragmentation is essential to the origin of the stellar IMF.

https://iopscience.iop.org/article/10.1086/341790/pdf

The Stellar Initial Mass Function from Turbulent Fragmentation

The morphology and kinematics of molecular clouds (MCs) are best explained as the consequence of super--sonic turbulence. Super--sonic turbulence fragments MCs into dense sheets, filaments and cores and large low density ``voids'', via the action of highly radiative shocks. We refer to this process as "turbulent fragmentation". In this work we derive the mass distribution of gravitationally unstable cores generated by the process of turbulent fragmentation. The mass distribution above one solar mass depends primarily on the power spectrum of the turbulent flow and on the jump conditions for isothermal shocks in a magnetized gas. For a power spectrum index \beta=-1.74, consistent with Larson's velocity dispersion--size relation as well as with new numerical and analytic results on super--sonic turbulence, we obtain a power law mass distribution of dense cores with a slope equal to 3/(4-\beta) = 1.33, consistent with the slope of the stellar IMF. Below one solar mass, the mass distribution flattens and turns around at a fraction of a solar mass, as observed for the stellar IMF in a number of stellar clusters, because only the densest cores are gravitationally unstable. The mass distribution at low masses is determined by the probability distribution of the gas density, which is known to be approximately Log--Normal for an isothermal turbulent gas. The intermittent nature of the turbulent density distribution is thus responsible for the existence of a significant number of small collapsing cores, even of sub--stellar mass. Since turbulent fragmentation is unavoidable in super--sonically turbulent molecular clouds, and given the success of the present model in predicting the observed shape of the stellar IMF, we conclude that turbulent fragmentation is essential to the origin of the stellar IMF.

The Stellar IMF from Turbulent Fragmentation

This work presents a new physical model of the star formation rate (SFR), verified with an unprecedented set of large numerical simulations of driven, supersonic, self-gravitating, magneto-hydrodynamic (MHD) turbulence, where collapsing cores are captured with accreting sink particles. The model depends on the relative importance of gravitational, turbulent, magnetic, and thermal energies, expressed through the virial parameter, alpha_vir, the rms sonic Mach number, M_S,0, and the ratio of mean gas pressure to mean magnetic pressure, beta_0. The SFR is predicted to decrease with increasing alpha_vir (stronger turbulence relative to gravity), to increase with increasing M_S,0 (for constant values of alpha_vir), and to depend weakly on beta_0 for values typical of star forming regions (M_S,0 ~ 4-20 and beta_0 ~ 1-20). In the unrealistic limit of beta_0 -> infinity, that is in the complete absence of a magnetic field, the SFR increases approximately by a factor of three, which shows the importance of magnetic fields in the star formation process, even when they are relatively weak (super-Alfvenic turbulence). In this non-magnetized limit, our definition of the critical density for star formation has the same dependence on alpha_vir, and almost the same dependence on M_S,0, as in the model of Krumholz and McKee, although our physical derivation does not rely on the concepts of local turbulent pressure and sonic scale. However, our model predicts a different dependence of the SFR on alpha_vir and M_S,0 than the model of Krumholz and McKee. The star-formation simulations used to test the model result in an approximately constant SFR, after an initial transient phase. Both the value of the SFR and its dependence on the virial parameter found in the simulations are shown to agree very well with the theoretical predictions.

The Star Formation Rate of Supersonic MHD Turbulence

The Padoan and Nordlund model of the stellar initial mass function (IMF) is derived from low-order statistics of supersonic turbulence, neglecting gravity (e.g., gravitational fragmentation, accretion, and merging). In this work, the predictions of that model are tested using the largest numerical experiments of supersonic hydrodynamic (HD) and magnetohydrodynamic (MHD) turbulence to date (~10003 computational zones) and three different codes (Enzo, Zeus, and the Stagger code). The model predicts a power-law distribution for large masses, related to the turbulence-energy power-spectrum slope and the shock-jump conditions. This power-law mass distribution is confirmed by the numerical experiments. The model also predicts a sharp difference between the HD and MHD regimes, which is recovered in the experiments as well, implying that the magnetic field, even below energy equipartition on the large scale, is a crucial component of the process of turbulent fragmentation. These results suggest that the stellar IMF of primordial stars may differ from that in later epochs of star formation, due to differences in both gas temperature and magnetic field strength. In particular, we find that the IMF of primordial stars born in turbulent clouds may be narrowly peaked around a mass of order 10 M☉, as long as the column density of such clouds is not much in excess of 1022 cm-2.

https://iopscience.iop.org/article/10.1086/516623/pdf

Two Regimes of Turbulent Fragmentation and the Stellar Initial Mass Function from Primordial to Present-Day Star Formation

Lada et al (1994) have described a method for studying the distribution of dust in dark clouds using infrared imaging surveys In particular they show that the method provides some information about the structure of the gas (dust) on scales smaller than their resolution 
In the present work we clarify the nature of the information provided by their method 
We show that: 
1) the 3D density field of the gas is well described by a Log-Normal distribution down to very small scales; 
2) the power spectrum and the standard deviation of the 3D density field can be constrained; 
3) the origin of such a structure of the density field is likely to be the supersonic turbulence in the gas 
In fact we find a qualitative and quantitative agreement between the predictions based on recent numerical simulations of supersonic turbulence (Nordlund and Padoan 1996; Padoan, Nordlund and Jones 1996) and the constraints given by the infrared dust extinction measurements

Aake Nordlund

Papers

The Stellar Initial Mass Function from Turbulent Fragmentation

The Stellar IMF from Turbulent Fragmentation

The Star Formation Rate of Supersonic MHD Turbulence

Two Regimes of Turbulent Fragmentation and the Stellar Initial Mass Function from Primordial to Present-Day Star Formation

Supersonic turbulence in the ISM: stellar extinction determinations as probes of the structure and dynamics of dark clouds